The Bogoliubov analytical results [1]} for elementary excitations in the Bose gas are very important for understanding the excitation spectrum in phonon region. Nevertheless the Bogoliubov's theory can be applied only to dilute Bose systems.
Feynman has found a relation between the energy spectrum of the elementary excitations and the structure factor [2]}, [3]} that verifies Landau's phenomenological dispersion relation. We note that the Feynman's relation is correct only for small enough momenta when the excitations are phonons. In a “roton-like" region it is only an approximation of a real situation.
Feynman has also proposed a model of the roton excitation as localised vortex ring [4]} with a characteristic size of the order of the mean atomic distance in liquid \({\rm ^4He}\) . A number of methods have been also suggested for applications to quantum Bose liquids. For a review of quantum fluid theories also see the references [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}.
